Competing magnetic ground states and their coupling to the crystal lattice in CuFe₂Ge₂

Abstract

Identifying and characterizing systems with coupled and competing interactions is central to the development of physical models that can accurately describe and predict emergent behavior in condensed matter systems. This work demonstrates that the metallic compound CuFe₂Ge₂ has competing magnetic ground states, which are shown to be strongly coupled to the lattice and easily manipulated using temperature and applied magnetic fields. Temperature-dependent magnetization M measurements reveal a ferromagnetic-like onset at 228 (1) K and a broad maximum in M near 180 K. Powder neutron diffraction confirms antiferromagnetic ordering below T_N ≈ 175 K, and an incommensurate spin density wave is observed below ≈125 K. Coupled with the small refined moments (0.5–1 μ_B/Fe), this provides a picture of itinerant magnetism in CuFe₂Ge₂. The neutron diffraction data also reveal a coexistence of two magnetic phases that further highlights the near-degeneracy of various magnetic states. These results demonstrate that the ground state in CuFe₂Ge₂ can be easily manipulated by external forces, making it of particular interest for doping, pressure, and further theoretical studies.

Introduction

Systems with a strong coupling between magnetism, the crystal lattice, and itinerant electrons often display interesting physics. When such systems have several nearly-degenerate ground states, complex emergent behavior can be observed, such as unconventional superconductivity¹. The behavior of systems with nearly-degenerate ground states can typically be tuned using external forces (pressure, magnetic field) or chemical manipulation, and identifying the so-called parent compounds that are characterized by magnetic and/or structural instabilities is one focus of science-driven synthesis.

CuFe₂Ge₂ has been predicted to possess an antiferromagnetic (AFM) ground state, with multiple magnetic configurations close in energy². There have been no experimental investigations of its physical properties. The theoretical calculations revealed multiple bands with a high density of Fe d states at the Fermi level, suggesting an itinerant character of the magnetism. The associated sheet-like structures of the Fermi surface provide nesting instabilities and thus promote the AFM ground state². Based on these calculations, Cu is anticipated to be non-magnetic.

CuFe₂Ge₂ is an intermetallic compound possessing an orthorhombic crystal structure (space group 51, Pmma)³. As shown in Fig. 1, it is characterized by sawtooth chains of Fe that lie within the ac plane. Fe(1) forms the centerline of the sawtooth chain running along the a-axis; it is coordinated by a distorted octahedra of Ge. These Fe(1) positions are separated by 2.49Å at room temperature, which is nearly identical to the nearest neighbor in metallic Fe (bcc) at room temperature (2.50 Å)⁴. Fe(1)-Fe(2) bonds are ≈2.66 Å, forming a sawtooth chain of isosceles triangles. The Fe(2)-Fe(2) distance between sawtooth chains along the c-axis is 3.25 Å, though bonds with copper link the chains. Thus, in spite of the three-dimensional crystal structure, the electronic and magnetic properties may be expected to possess significant anisotropy.

Figure 1

(a,b) Crystal structure of CuFe₂Ge₂ highlighting the sawtooth chains of Fe atoms residing in the (ac) plane. (c) Powder x-ray diffraction data of CuFe₂Ge₂ at room temperature. The calculated curve is from Rietveld refinement, with atomic positions of Cu at 0, 0, 0; Ge(1) at , 0, 0.380(1); Ge(2) at ,  , 0.198(1); Fe(1) at 0, ,  ; and Fe(2) at ,  , 0.155(1).

This study reports the synthesis and characterization of polycrystalline CuFe₂Ge₂. Neutron powder diffraction reveals an incommensurate spin density wave between the base temperature of 4 K and 125 K, while commensurate AFM order is observed below the Néel temperature of ≈175 K down to 100 K. Interestingly, a coexistence of these two magnetic phases is observed in the crossover region. Magnetization measurements reveal a clear increase in the magnetization below T₀ ≈ 228 K, and a small ferromagnetic component remains upon cooling into the AFM state. These measurements also demonstrate an evolution of the magnetization as a function of applied field, with relatively small changes in the field strongly affecting the observed behavior. In addition, the magnetism is found to be strongly coupled to the lattice, as demonstrated by anisotropic thermal expansion and a strong change in the c lattice parameter at the magnetic phase transitions. In total, these results demonstrate that CuFe₂Ge₂ has complex magnetism resulting from nearly-degenerate states that can be easily manipulated. The application of external forces or chemical doping will likely produce additional emergent states in CuFe₂Ge₂, potentially including superconductivity.

Results

Zero-Field Limit of Magnetism

Figure 2 provides an overview of the temperature-dependent data utilized to characterize the magnetic phase transitions in CuFe₂Ge₂. The temperature dependence of the magnetization M was found to depend strongly on the applied field H. To probe the magnetic ground state, data were collected while cooling in a very small field of H = 2 Oe; the data are plotted as M/H, which equals the magnetic susceptibility χ when M is linear with H. As observed in Fig. 2a, M increases sharply upon cooling below T₀ ≈ 228 K, which appears similar to the onset of ferromagnetic order. However, the magnetization reaches a maximum near 181 K and then decreases upon cooling until essentially plateauing below ≈120 K. The decrease in M upon cooling suggests antiferromagnetic (AFM) order, though the shape of M(T) is unusual and it is difficult to define a Néel temperature T_N from these data. Therefore, we define the maximum in M(T) as T ^*, and obtain T_N from neutron diffraction data that clearly demonstrate AFM order. The shape of M(T) indicates unusual magnetism or competition between different magnetic states. The magnetic susceptibility has a large temperature-independent contribution above ≈300 K, and high T susceptibility measurements are necessary to further probe the potential role of localized moments in the non-magnetically-ordered state.

Figure 2

Magnetic phase transitions in CuFe₂Ge₂ illustrated using (a) temperature-dependent magnetization and (b–d) neutron diffraction data. (b) Intensity at magnetic and nuclear Bragg reflections as a function of temperature; data for the reflection were collected using a fixed |Q| = 1.21 Å = 26.9 degrees 2θ, whereas the other intensities come from peak fitting. (c) Magnetic Bragg reflection with propagation vector and (d) emergence of additional magnetic reflections at 125 K, which come from an incommensurate spin structure with propagation vector . The magnetic Bragg reflections are indexed.

Neutron diffraction data demonstrate AFM order below T_N ≈ 175 K, as well as the transition from a commensurate (AFM − C) to incommensurate (AFM − IC) spin structure upon cooling below ≈125 K. These transitions are shown in Fig. 2b, where the intensities of several neutron diffraction reflections are plotted as a function of temperature. AFM order is demonstrated by intensity at the and Bragg reflections. This magnetic scattering is described by the propagation vector , which indicates the moments are antiferromagnetically-aligned along the b-axis. The intensity of magnetic Bragg peaks from AFM − C starts to increase near 175(5) K, reaches a maximum near 135 K, and then decreases to near background levels at the base temperature of 4 K. At 125 K, a set of magnetic Bragg reflections from AFM − IC are observed (such as those at 26 and 28.4 deg. 2θ in Fig. 2d). These magnetic reflections are indexed to the incommensurate propagation vector , and their intensity increases upon cooling to the base temperature of 4 K. In Fig. 2d, the two outermost peaks move apart upon cooling, which corresponds to an increase in x (a decrease in the real-space sinusoidal period). Refinement yields x = 0.07765 (125 K), 0.08723 (100 K), 0.1044 (70 K) and 0.117 (4 K). A clear coexistence of AFM − C and AFM − IC is observed between approximately 70 and 125 K, indicating a competition between magnetic phases in this region. Note that at 200 K neutron diffraction intensity is not detected at locations of the peaks now identified as magnetic contributions (see Supplementary Information), and x-ray diffraction does not reveal intensity at these locations.

These results demonstrate the competition between two or more magnetic structures in CuFe₂Ge₂, and the transition between a commensurate and an incommensurate magnetic state may have implications regarding the movement of electronic bands as a function of temperature. In elemental chromium, the prototype itinerant AFM, the spin density wave is incommensurate and a spin-flop is observed at about 0.4T_N. The SDW in chromium is driven by electronic nesting, and an imbalance in electron and hole pockets leads to the incommensurate structure. Due to the electronic origin, the SDW can be easily manipulated by doping. For instance, the addition of Mn leads to the stabilization of commensurate magnetic order, and for some small concentrations of Mn a transition between commensurate and incommensurate SDWs is observed upon cooling⁵. These transitions in Cr-Mn have been considered within the framework of defects and electronic damping⁶. It would be interesting to see if the transition between AFM − C and AFM − IC in CuFe₂Ge₂ were influenced by anti-site defects or other intrinsic dopants. Such chemical disorder may be difficult to identify in CuFe₂Ge₂ due to the similar atomic masses. However, the changes with annealing T that are discussed in the Supplementary Information may allow for future experiments that examine the influence of disorder on the observed transitions.

At 135 K, the neutron data can be fit to a spin structure that is antiferromagnetic along b, with chains of Fe(1) atoms ferromagnetically coupled along a and antiferromagnetically coupled with Fe(2). This corresponds to the irreducible representation Γ4 (see Methods). The magnetic structure obtained for the AFM − C phase is consistent with the ‘AF-G’ ground state predicted by Shanavas and Singh². Unfortunately, the neutron data can be described equally-well when the antiferromagnetically-aligned Fe moments point along a or along c (or a combination of the two). Our first principles calculations find that the c-axis is the preferred orientation of the moments (see Methods for details), and thus the Fe moments were constrained to point along c. The corresponding magnetic structures are shown in Fig. 3, and we emphasize that these are the simplest models that describe the data. The refined moments are 0.36(10) μ_B/Fe(1) and 0.55(10) μ_B/Fe(2) at 135 K. Small moments can be expected near T_N, though this observation also lends support to the hypothesis that the magnetism is itinerant in CuFe₂Ge₂. Note that the system is metallic in regards to the temperature-dependence of its electrical resistivity (see Supplementary Information).

Figure 3

Comparison of nuclear and magnetic structures in CuFe₂Ge₂.

(a) Two units cells of the nuclear structure, with atomic positions labeled. (b,c) The magnetic models utilized at (b) 135 K and (c) 4 K, with only the Fe positions and moments shown. The spin structure shown in (c) is a commensurate structure with propagation vector , which closely represents the observed incommensurate structure with propagation vector at 4 K.

Given the theoretical prediction of itinerant magnetism, and the experimentally confirmed reduced magnetic moment, the incommensurate phase AFM − IC is treated with a spin density wave (SDW) model, though a helical structure would also describe the scattering. The proposed structure for AFM − IC shown in the lower portion of Fig. 3 is represented by a commensurate structure with propagation vector ; the measured propagation vector is at 4 K. Refinement of the 4 K data yields moments of 1.0(1) μ_B/Fe(1) and 0.71(10) μ_B/Fe(2). These represent the maximum moments because they vary as a function of z. Our first principles calculations (using LDA) obtained ±1.25μ_B/Fe(1) and ±0.90μ_B/Fe(2) for the magnetic ground state AF0 (see Methods); note that the symmetry of AF0 is consistent with AFM − C. Reasonable agreement between theory and experiment is observed, especially considering that the calculation is being performed in a different magnetic symmetry than the experimentally-observed incommensurate structure. For instance, the agreement here is better than that in the Fe-based superconductors, where theory significantly overestimated the Fe moments despite utilizing the experimentally-observed magnetic structures⁷.

Coupling of Magnetism and the Crystalline Lattice

Shanavas and Singh reported first principles calculations that demonstrated various magnetic symmetries/configurations are within 20 meV of each other². Our calculations are consistent with this, and thus the observed properties may be understood as a consequence of nearly-degenerate ground states. However, the presence of nearly-degenerate ground states does not necessarily explain the evolution of magnetism, especially in a system that orders at fairly high T. Therefore, an additional energy or order parameter must be important in driving the magnetic transitions.

One driving force for stabilizing a particular magnetic phase can be magnetoelastic coupling. To probe this behavior, x-ray diffraction data were collected as a function of temperature (see Fig. 4). Consideration of magnetoelastic effects is greatly aided by comparison with a non-magnetic analogue. For this system, we found that a sample of nominal composition CuFeCoGe₂ provides a non-magnetically-ordered baseline to consider. Magnetization data show that this composition appears paramagnetic down to at least 5 K, though a small anomaly was observed near 150 K (see Supplementary Information for powder x-ray diffraction and magnetization data). The x-ray diffraction data for CuFeCoGe₂ are well-described using the CuFe₂Ge₂ model, and at room temperature the lattice parameters are a = 4.9966(1) Å, b = 3.9262(1) Å, and c = 6.7162(1) Å. Compared to a = 4.9765(1) Å, b = 3.9718(1) Å, and c = 6.7834(1) Å for CuFe₂Ge₂ at 300 K, these results show that cobalt incorporation increases a while b and c decrease. The unit cell volume of CuFeCoGe₂ is nearly 2% smaller than that of CuFe₂Ge₂.

Figure 4

(a) Evolution of relative lattice parameters as a function of T in CuFe₂Ge₂. (b) The refined c lattice parameter in CuFe₂Ge₂. Data for two samples are shown, with the open symbols representing the sample that was also used for neutron diffraction. (c) Relative lattice parameters versus T in the doped, non-magnetically-ordered composition CuFeCoGe₂ demonstrating a smooth contraction with decreasing T in the absence of magnetic order. The inset shows the relative change in unit cell volumes, with red crosses for CuFe₂Ge₂ and open squares for CuFeCoGe₂. In (a,c) the error bars are smaller than the data markers.

The thermal expansion is found to be anisotropic in CuFe₂Ge₂, with c changing much less than a or b (Fig. 4a). As shown in Fig. 4b, the c lattice parameter has a non-monotonic T-dependence and responds strongly to the magnetic order near T_N ≈ 175 K and again below T ≈ 120 K (where the incommensurate structure begins to dominate). This is not the case for CuFeCoGe₂, where the thermal expansion is much more isotropic and monotonic (Fig. 4c). As seen in the inset of Fig. 4c, the relative changes in the unit cell volumes are nearly identical for the doped and undoped samples. This shows that expansion along a and b are enhanced for CuFe₂Ge₂ in response to the stiff behavior along c. It is worth noting that the sawtooth chains of Fe exist in the ac plane, running along a with the shortest distance between chains being along c. These results clearly demonstrate a strong coupling between the lattice and the magnetism in CuFe₂Ge₂, and in particular they suggest that the application of anisotropic strain would likely have a significant impact on the ground state in CuFe₂Ge₂.

First principles calculations were performed to further investigate the role of magnetoelastic coupling in CuFe₂Ge₂. An overview of the results is provided here and additional details are given in the Supplementary Information. In the parent phases of the Fe-based superconducting (SC) systems, where strong magnetoelastic coupling is observed, the calculated moments were found to vary strongly with the Fe-As bond length^7,8,9,10. To check for similar behavior in CuFe₂Ge₂, the moments were calculated as a function of the z coordinate of Fe(2). As z increases from its equilibrium position, moving Fe(2) toward Fe(1), the moment on Fe(1) increases slightly and the moment on Fe(2) decreases. A relative increase in the distance between Fe(1) and Fe(2) causes very little change in the moments. These different dependencies on z illustrate a difference between CuFe₂Ge₂ and the Fe-based superconductors, and are one manifestation of the relatively complex magnetic interactions and competing ground states in CuFe₂Ge₂. In a complementary calculation, the influence of the imposed magnetic symmetry on this z coordinate was investigated by relaxing the atomic positions for different magnetic states. These calculations found that z increases by 7.5% in the ferromagnetic state relative to the AFM ground state (bringing Fe(1) and Fe(2) closer together). The z of Fe(2) only changed by 1.3% between the AFM ground state and the non-magnetic state, though. Refinement results from x-ray diffraction data did not reveal any significant changes in atomic positions as a function of T as shown in the Supplementary Information.

Field Dependence of Magnetism

The application of a magnetic field changes the shape of M(T), as shown in Fig. 5(a,b). In Fig. 5a, a single maximum defined as T ^* can be observed, but the application of fields H > 2 kOe leads to the presence of two distinct maxima at T₁ and T₂ (see Fig. 5b). This is a relatively small field to produce such a large change in the magnetism at these temperatures. Specifically, because the moments are small the total magnetic energy is small; for a moment of 1μ_B the magnetic energy is equivalent to 0.67 K for every 10 kOe applied. As such, it is surprising that the magnetism can be changed by such a small field and these results may also have implications for the response of the lattice to an applied field.

Figure 5

Influence of applied magnetic field on the temperature dependence of magnetization and specific heat.

(a) M(T)/H is shown for small fields where a single maximum at T ^* is observed, and zero-field cooled data (ZFC) are shown for 100 Oe. (b) The evolution of M(T) with applied fields revealing critical temperatures T₁ and T₂ for  2 kOe. (c) Phase diagram of the critical temperatures from M(T) curves, with error bars shown for a few points. These temperatures are poorly defined in the shaded region where all three are observed. Data for multiple samples are shown for comparison. (d) Isothermal magnetization curves at various temperatures highlighting the low-field data that demonstrate an increase in the small ferromagnetic component with decreasing T; the inset shows M(H) is mostly linear at 2 and 250 K. (e) Specific heat capacity data with low-T fit shown in the inset. (f) The anomaly in the specific heat capacity near 170 K is found to be unchanged by applied fields of 100 Oe and 50 kOe; the derivative is plotted on the right-side axis.

The phase diagram demonstrating the evolution of T ^*, T₁, and T₂ with applied field is shown in Fig. 5c. T ^* decreases abruptly with applied field; similar behavior is typically observed for the Néel temperature of an antiferromagnetic transition, although usually with a much weaker field dependence. A region between approximately 700 and 2000 Oe appears to have a coexistence of all three characteristic temperatures, and it is difficult to define T ^*, T₁, and T₂ in this range (though estimates are shown). In addition, the M(T) curves have rather broad features/maxima, making the definitions of critical temperatures especially difficult for intermediate fields. Outside of this region, the typical error for these critical temperatures is about 3 K for a given sample. Data for three different samples are included in Fig. 5c (represented by closed circles, open circles, and ×). The different samples possessed similar behavior, though a notable difference for T₂ was observed in the sample used for neutron diffraction (open circles). This may suggest that sample possessed a different amount of atomic disorder or interstitial occupancy. Importantly, however, the behavior of T ^* is consistent among all samples, indicating that the low-field behavior of all samples is consistent and thus the neutron diffraction results should be representative of this ground state. While not shown, T₀ associated with the onset of magnetization increases with increasing H, as expected for ferromagnetic order.

Isothermal magnetization measurements at various temperatures are shown in Fig. 5d, where the main panel emphasizes the small ferromagnetic contribution observed at low fields. As shown in the inset, the M(H) curves are nearly linear for all temperatures, consistent with a paramagnetic state or antiferromagnetic order. A small ferromagnetic component is observed starting at 225 K, consistent with the sharp rise in magnetization near T₀ ≈ 228 K; the remanent moment and coercivity increase with decreasing T. Since neutron diffraction clearly shows this to be antiferromagnetically ordered below ≈175 K, we speculate that the ferromagnetic component (the remanence, coercivity) arises due to either a canting of the antiferromagnetically-aligned moments or from the magnetization of interstitial or anti-site Fe species. The FM component increases as the annealing temperature increases (and crystallinity decreases), but the temperature-dependence remains the same. This suggests that the degree of canting or the number of FM sites is dependent on the concentration of point defects (site disorder, vacancies, interstitials). As such, it does not appear that the FM component originates from an impurity phase.

Specific heat capacity data are shown in Fig. 5e,f. The anomaly at T_N is subtle, with an onset near 170 K as shown by the derivative curve in Fig. 5f. Interestingly, the C_P anomaly does not change in applied fields, which is fairly unusual for magnetic phase transitions and is especially surprising for one that is easily influenced by an applied field. The lack of an anomaly in C_P near T₂ ≈ 125 K for H = 50 kOe suggests that the feature in M(T) at T₂ is related to a reconfiguration of the moments. That is, if the magnetic entropy is quenched at T_N or T₁ then the anomaly in M(T) at T₂ should not have a corresponding anomaly in C_P. Therefore, T₂ may be related to a cross-over between the commensurate and incommensurate antiferromagnetic structures, or perhaps it is associated with a reorientation of the moments (between crystallographic axes). In either case, the emergence of T₂ clearly demonstrates the sensitivity of magnetism in CuFe₂Ge₂ to applied fields.

The inset of Fig. 5e highlights the low T data, which were fit to C_P/T = γ + βT ², where γ is the Sommerfeld coefficient associated with electronic contributions and β represents the Debye phonon contribution; note that we are neglecting magnetic contributions. A relatively large Sommerfeld coefficient of γ = 25.2 mJ/mol/K² was obtained; from β a Debye temperature of 385 K was calculated. The large value of γ is likely associated with the presence of Fe d states near the Fermi level. Using first principles calculations, Shanavas and Singh obtained a bare (non-magnetically ordered) Sommerfeld coefficient of 18.6 mJ/mol/K², suggesting that the Fermi surface naturally has a large density of states that promotes itinerant magnetism². For comparison, γ = 16 mJ/mol/K² in BaFe₂As₂¹¹, which is a parent compound for Fe-based superconductors and has a spin density wave transition near 140 K.

Discussion and Summary

The data reveal that CuFe₂Ge₂ has competing magnetic states with dominant antiferromagnetic order at low T. Neutron diffraction demonstrates AFM order below ≈175 K, and an incommensurate spin structure is observed below ≈125 K. The magnetization has an unusual temperature dependence with a broad maximum near T_N, and the results suggest proximity to ferromagnetic ordering. A weak ferromagnetic contribution is evident in the magnetization data below ≈228 K and divergence of field-cooled and zero-field-cooled data is also observed. Comparison of the zero-/low-field results to the field-dependent data provides some insights into the nature of the magnetic ordering. For instance, the C_P anomaly occurs roughly 9 K below the maximum of M(T) observed for an applied field of 2 Oe (the largest T ^* in Fig. 5d). The C_P anomaly does not change with applied field, while the maximum in M(T) is strongly suppressed with increasing H. These discrepancies demonstrate the difficulty in defining T_N based on the M(T) data due to the FM-like onset and the unusual shape of M(T). In a more general sense, though, these results may point to a gradual and complex evolution of the magnetism. For instance, the onset of magnetization may be associated with local ordering or canted-antiferromagnetic ordering of one Fe sublattice; a complex ferrimagnetic arrangement may also exist. Local probe measurements, such as Mössbauer spectroscopy, may prove particularly insightful. Measurements on single crystals would certainly prove useful. However, due to the decomposition/disordering of the phase above 630°C, care must be taken during crystal growth. In this regard, a measurement of the magnetization appears to be a useful probe of sample quality; smaller M and smaller ferromagnetic-contributions at low T seem to indicate higher sample quality.

In summary, CuFe₂Ge₂ possesses complex magnetism with a magnetic structure that evolves as a function of temperature and applied field. In addition, cobalt doping has been shown to suppress the magnetic order. Experimental and theoretical results suggest magneto-structural effects play an important role in determining the magnetic ground state. These results also clearly demonstrate that the magnetism in CuFe₂Ge₂ has a strong itinerant character, as illustrated by neutron diffraction data revealing small Fe moments and an incommensurate spin density wave below ≈125 K. Unlike the Fe-based superconductors, a competition between various magnetic ground states leads to a complex magnetic structure and an unusual temperature-dependence of the magnetization. Given these observations, studies into the pressure-dependence and doping-dependence of the physical properties in CuFe₂Ge₂ will likely demonstrate the ability to tune between different magnetic states, and could potentially result in other emergent behavior such as superconductivity.

Methods

Synthesis and Characterization

Polycrystalline samples of CuFe₂Ge₂ were prepared by arc melting high-purity elements on a cooled-copper hearth, followed by grinding, cold-pressing and annealing. The annealing temperature was found to be critical, and the samples reported upon here were annealed at 600°C for 7 d. As discussed in the Supplementary Information, significant degradation of the crystallinity is observed when the samples are annealed at 700 °C as opposed to 600 °C. This is likely due to decomposition and/or disordering of the phase above approximately 615–635 °C.

Phase purity was investigated using a Panalytical X’PERT PRO diffractometer with monochromatic copper K_α,1 radiation, and temperature-dependent structural information was obtained by employing an Oxford closed cycle cryostat. A representative Rietveld refinement is shown in Fig. 1c. The samples were observed to be phase pure to within ≈1%, which is roughly the limit of laboratory x-ray diffraction. Rietveld refinement using the published orthorhombic structure yielded a = 4.9765 (1) Å, b = 3.9718 (1) Å, and c = 6.7834 (1) Å at 300 K. For the data in Fig. 1c, the standard refinement quantifiers¹² are R_p = 3.38, R_wp = 4.29, and χ² = 1.36.

Physical property measurements were completed in Quantum Design systems (Physical and Magnetic Property Measurement systems) using standard practices for sample mounting and contacting. Ag epoxy was employed for contacts during four point electrical and thermal transport measurements, and N-grease and H-grease were used for specific heat measurements at T < 200 and T > 200 K, respectively.

Neutron Diffraction Data Collection and Analysis

Neutron diffraction data were collected on the powder neutron diffractometer HB2A at the High Flux Isotope Reactor in Oak Ridge National Laboratory. A sample of ≈8 g was held in an Al can and data were obtained using a neutron wavelength of 2.4123 Å. This set-up is optimized for measuring small magnetic peaks, but limits the structural refinement details. Neutron and x-ray diffraction data were refined using the program FullProf¹² and the representational analysis was performed with SARAh¹³.

Representational group analysis was performed to obtain symmetry-allowed magnetic structures from the neutron powder diffraction data at 135 K and 4 K (see Supplementary Information). This yields three common irreducible representations (IRs) for the Fe sites (Γ_mag = Γ₂ + Γ₄ + Γ₈). The observation of a reflection implies that the magnetic moment has a component perpendicular to the b axis and this therefore suggests the IR description composed of b-axis parallel spins can be discarded (Γ₂). Representational analysis limits the number of symmetry allowed magnetic structures, however within these the various combinations of couplings between the different Fe sites (basis vectors) necessitates further information to obtain a robust magnetic structure. For these data, it is not possible with the limited number of peaks to distinguish between the IRs Γ₄ and Γ₈. As such, first principles calculations were utilized to provide insight into the preferred orientation of the magnetic moments.

It is very possible that, particularly for the incommensurate SDW, the orientation of the moments is not along a high-symmetry direction. The data have been refined using a modulated spin structure (sinusoidal modulation of the moment), which produces zero moment on some Fe sites. Note that the magnetic scattering can also be described using a helical spin structure and polarized measurements on single crystals would be required to differentiate between these two types of incommensurate structures. Given the apparent itinerant nature of this magnetism, a SDW model seems more appropriate than a helical one. The data above 130 K can be refined using the AFM − IC structure with very small x. However, the simplest magnetic structure that describes the data was utilized, yielding the magnetic structure in Fig. 3a. The coexistence of the two magnetic phases at 125 K may also suggest that the structures at 4 K and 135 K are fundamentally different, as one may expect a continuous evolution of x with T if the structure at 135 K were a nearly-commensurate limit of the incommensurate structure (meaning there would not be a region of coexistence).

Complete neutron diffraction data sets were collected at several temperatures to investigate the nature of the observed magnetic transitions. These data and corresponding Rietveld refinements are shown in the Supplementary Information. At 200 K, the neutron diffraction data are well described by a non-magnetic model that is consistent with the structure used for x-ray diffraction data. Note that the magnetization data suggest a ferromagnetic-like onset at 228 K. Below T_N, the observed magnetic scattering is not particularly strong relative to the nuclear Bragg peaks, and thus it is beyond the limits of the measurement to observe a small ferromagnetic component in this material.

First Principles Calculations

The electronic structure calculations were performed using the all-electron planewave code WIEN2K¹⁴. All results presented in this paper were performed using the experimental lattice parameters, with internal coordinates not dictated by symmetry optimized until the ionic forces were less than 2 mRyd/Bohr. The calculations of magnetostructural properties employed the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof¹⁵ while the calculations of easy axis orientation used the local density approximation (LDA). For the GGA calculations an RK_max of 7.0 and the augmented plane wave basis was used, where RK_max is the product of the smallest LAPW sphere radius and the largest planewave expansion vector. For the LDA calculations an RK_max of 8.0, and the linearized augmented plane-wave basis, were used. For all calculations, sphere radii of 2.09 Bohr for Ge, 2.20 for Fe and 2.24 for Cu were employed. For all self-consistent calculations, approximately 1000 k-points in the full Brillouin zone were used. It is difficult to perform first principles calculations on non-collinear magnetic structures, and thus we have limited our analysis to collinear configurations.

The magnetic ground state is termed AF0 and is consistent with the experimental data at 135 K; AF0 is equivalent to the AF-G ground state found in the previous work². The first excited state is termed AF1. In both antiferromagnetic states the Fe(1)-Fe(1) nearest neighbor (nn) chains (along the a-axis) are aligned ferromagnetically and the next nearest-neighbor (nnn) Fe(2) is anti-aligned to Fe(1); in AF1 the nnnn (along the b-axis) Fe(1) is aligned to the base cell Fe(1) and in AF0 this nnnn Fe1 is antialigned. We could not stabilize an ab-plane checkerboard state, as was studied in the previous work; calculations initialized in this pattern converged to the AF0 state.

Magnetoelastic effects were probed using AF0, AF1, a non-magnetic and a ferromagnetic configuration. The experimental lattice parameters at T = 220 K were imposed, and then the internal coordinates that are not dictated by symmetry were optimized. The greatest change was observed for z of Fe(2), and the obtained values are placed in the Supplementary Information along with the calculated energies. The non-magnetic state has an optimized z for Fe(2) of 0.1541, and the AFM ground state has a similar z of 0.151. This is in fair agreement with the experimentally-obtained z of 0.155(1) between 60 and 300 K. In addition, the moments within AF0 were calculated as a function of the Fe(2) z after fixing all other crystallographic parameters (coordinates for Ge(1) and Ge(2) were optimized first). A plot summarizing these results is shown in the Supplementary Information.

In order to guide the refinement of the neutron diffraction data, very careful calculations of the total energy for Fe moments oriented along the three different axes were conducted within the AF0 ground state with the incorporation of spin-orbit coupling. Note that the magnetic pattern itself - the relative orientation of moments on different Fe sites - is unchanged for these calculations. What we study here is the coupling of the moments to the crystalline structure, or effectively the magnetic anisotropy. As is well known, this anisotropy derives from spin-orbit coupling, which is expected to be weak here as all elements are relatively light (one recalls the effective Z⁴ dependence¹⁶ of spin-orbit coupling). Hence these energy differences are expected to be very small. Thus, a minimum of 10,000 k-points were used for these total energy calculations. Increasing the number of k-points to 30,000 changed energy differences by less than 1%. These calculations were done using the experimental lattice parameters at 60 K and 220 K, and little difference was observed; the 60 K results are reported. The moment orientation parallel (and anti-parallel) to the c-axis is predicted to be the ground state, being favored by 0.117 meV/Fe relative to placing the spins along the a-axis and 0.037 meV/Fe relative to spins along the b-axis. These anisotropies are in the typical range for non-cubic, 3d materials whose only magnetic elements are 3d elements; hcp Cobalt, for example exhibits a magnetic anisotropy of approximately 0.06 meV/Co¹⁷.